Robust and Pristine Top ological Dirac Semimetal Phase in Pressured Two-Dimensional Black Phosphorous

TL;DR

This study demonstrates that applying hydrostatic pressure to phosphorene can induce a pristine two-dimensional topological Dirac semimetal phase with symmetry-protected Dirac cones, offering a feasible experimental pathway for topological phase transition.

Contribution

The paper reveals that hydrostatic pressure, unlike electric fields or doping, can reliably induce a pristine 2D TDSM phase in phosphorene, clarifying the underlying mechanism.

Findings

Hydrostatic pressure activates topological phase transition in phosphorene.

The resulting TDSM state has a single pair of symmetry-protected Dirac cones.

Dirac points are robust under external perturbations if glide-plane symmetry is preserved.

Abstract

Very recently, in spite of various efforts in searching for two dimensional topological Dirac semimetals (2D TDSMs) in phosphorene, there remains a lack of experimentally efficient way to activate such phase transition and the underlying mechanism for the topological phase acquisition is still controversial. Here, from first-principles calculations in combination with a band-sorting technique based on k.p theory, a layer-pressure topological phase diagram is obtained and some of the controversies are clarified. We demonstrate that, compared with tuning by external electric-fields, strain or doping by adsorption, hydrostatic pressure can be an experimentally more feasible way to activate the topological phase transition for 2D TDSM acquisition in phosphorene. More importantly, the resultant TDSM state is a pristine phase possessing a single pair of symmetry-protected Dirac cones right at the Fermi level, in startling contrast to the pressured bulk black phosphorous where only a carrier-mixed Dirac state can be obtained. We corroborate that the Dirac points are robust under external perturbation as long as the glide-plane symmetry preserves. Our findings provide a means to realize 2D pristine TDSM in a more achievable manner, which could be crucial in the realization of controllable TDSM states in phosphorene and related 2D materials.

Equations

$$\delta ({\text{K}}_i)$$

$$(-1{)}^{\nu }$$

$$|\sum _i{[c_i^{n_1,J_1}(k)]}^{\ast }\cdot c_i^{n_2,J_2}(k+{\Delta }_k)|={\delta }_{n_1,n_2}\cdot {\delta }_{J_1,J_2}+O({\Delta }_k).$$

Full text

\alsoaffiliation

Department of Electronics and Tianjin Key Laboratory of Photo-Electronic Thin Film Device and Technology, Nankai University, Tianjin 300071, China \altaffiliationContributed equally to this work

\altaffiliationContributed equally to this work

\alsoaffiliationUniversity of Science and Technology of China, Hefei 230026, China

\alsoaffiliationUniversity of Science and Technology of China, Hefei 230026, China

\AbstractOn

{tocentry}

Two dimensional (2D) Dirac semimetals, in which Dirac points cross the Fermi level ($E_{\text{F}}$) being protected by nonsymmorphic crystal symmetries, was first proposed by Young and Kane in 2015 1. Very recent progress on the strain or electric-field modified phosphorene, as a representative of the rare candidates, has established a link between Dirac cones and the topological nature 2, 3, 4. Given the experimental discovery of three dimensional (3D) topological Dirac semimetals (Na3Bi 5, Cd3As2 6) and $\alpha$-Sn on InSb(111) substrate 7, it is natural to ask whether the 2D topological Dirac semimetals (TDSMs) can also be realized in an experimentally feasible manner.

A simple mechanism for a 2D TDSM phase acquisition has been proposed based on the Stark effect in phosphorene thin films by applying an external electric field 3, 8. However, applying an exceptionally giant electric field on such a system is difficult to realize experimentally (the value of the field required is $\sim$0.5 V/Å for a four-layer phosphorene 8). On the basis of the same mechanism, the experimentally observed Dirac semimetal state, from potassium doping of few-layer phosphorene, is actually an electron-doped TDSM 9. However, a pristine TDSM in 2D is highly desirable as it can be tuned to be topological insulators (TIs) or Weyl semimetals by explicit breaking of symmetries. In this regard, in-plane strain has been proposed as a possible means to induce Dirac cones in monolayer or bilayer phosphorene 10, 11, 12, whereas the critical strain (as large as $\sim$10% uniaxial strain or 5% biaxial strain based on the DFT-PBE level 11) is difficult to be experimentally realized, particularly for the biaxial strain in strong-anisotropic systems such as phosphorene. Furthermore, such large uniaxial/biaxial strains could substantially lower the local symmetry of the system, resulting in large distortion and structural instability due to the accumulative strain energy. Additionally, it remains conflicted in literature about the true phase (i.e., whether the resultant phase is a TI or a TDSM) in the strain-induced phase transition 10, 4, 2. All the above issues need to be solved or clarified, and examined by a more achievable way.

Apart from uniaxial/biaxial strains, the hydrostatic pressures, which to a larger extent preserve the crystal symmetry, and also could be free of the typical epitaxial-mismatch effect, have been proved as a powerful tool to deal with both fundamental and practical issues. Very recently, we have reported that bulk black phosphorous (BP) can convert from a normal insulator (NI) into a 3D Dirac semimetal under a certain hydrostatic pressure, both experimentally and theoretically, but only a carrier-mixed phase is acquired 13, 14. In experiments, hydrostatic pressures can also substantially modify the optical and vibrational properties of 2D systems like monolayer or few-layer MoS2 15, 16, 17, 18, 19, 20, WS2 21, and WSe2 22. Starting from this point, the few-layer phosphorene could be feasibly pressurized like other layered 2D systems. Therefore, in this paper, from first-principles methods in conjunction with band-sorting technique based on k$$\cdot$$p theory (to solve the above mentioned TI or TDSM conflict), we study the layer-pressure topological phase diagram of few-layer phosphorene. We discovered that hydrostatic pressures can energetically drive the 2D phosphorene from NI to TDSM phase, with the topological-phase-transitions (TPTs) depending on the number of layers. The resultant 2D TDSM state is characterized to be a pristine phase as a consequence of the reduced dimensional effect in the vertical direction. We demonstrate that the Dirac points are robust under external perturbation, including strain, pressure or electric field, as long as the glide-plane symmetry within each sublayer preserves.

In order to explore the evolutions of electronic structures of few-layer phosphorene under increasing hydrostatic pressures, we first determine their crystal structures (under each pressure) by DFT calculations. We employed the Vienna Ab initio Simulation Package (VASP) 23 with the projector augmented wave (PAW) method 24, 25. The interlayer vdW interaction is described by the optB88-vdW functional 26. A previous study on few-layer MoS2 has proposed an extraction method to obtain the 2D structures under pressures 27. Based on this method, their results reveal that the critical pressure for direct-to-indirect gap transition is $\sim$13 GPa, in qualitative agreement with the conclusion drawn from photoluminescence experimental measurement of $\sim$16 GPa 15. The extraction method is a reasonable and practical strategy to mimic the real pressurized procedure for layered systems, because it makes the sample feel the pressure from the other parts of the bulk system, which could serve as a good approximation to the inert pressure-transmitting medium in experiments. Meanwhile, the periodic potential field at the vdW boundary is not such strong that can change the geometric and electronic properties of the 2D BP system, as this is further verified by another method, namely, the medium method, in which the very realistic pressure-transmitting medium is considered. We find based on a set of comparison test, that basically the two method will yield a same structure for the 2D BP under a given pressure (see Table S2), suggesting that the extraction method is reliable to the cases in this study. As it is computationally much more expensive to consider the medium in each configuration (more than 800 atoms in the whole system), especially for the cases with the inclusion of spin-orbital coupling (SOC) effect, here, we chose the extraction method as an appropriate approach to conduct the study. Within the extraction method, in our study the structure of bulk BP was first optimized under a specified hydrostatic pressure, and then these structural parameters were used to construct the $n-$layer phosphorene, with a vacuum space of 20 Å along the $z$ direction. Based on the constructed 2D structures, the electronic properties is calculated by hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06) 28 calculations with the inclusion of SOC effect.

Few-layer BP is a layered material, in which each layer stacks together by van der Waals (vdW) interactions. Each layer consists of two sublayers and in turn forms a bulking honeycomb structure. The structural model of 2-layer phosphorene, as a representative for a finite $n$-layer phosphorene ($n$ denotes the number of layers), is shown in Figures 1a,b, where the directions of lattice vectors $a$ and $b$ are along the zigzag (the $y$ axis) and the armchair (the $x$ axis) directions, respectively. The 2D rectangular Brillouin zone of $n$-layer phosphorene is typified in Figure 1c.

With increasing pressure, as shown in Figure 1d, the lattice constant $b$ (armchair direction) and the interlayer spacing between two adjacent phosphorene layers ($d_{\text{int}}$) are significantly shortened (by 5.6% and 8.8% for $b$ and $d_{\text{int}}$ at 4.0 GPa), while $a$ (zigzag direction) and the sublayer distance within one single layer ($h$) remain almost unchanged. The thermodynamic and lattice-dynamic stabilities of few-layer phosphorene depend on their atomic and electronic properties, as well as the environment, such as pressure and temperature, etc. In the current work, the stabilities of few-layer phosphorene are in line with that of bulk BP. The thermodynamic stability of the bulk BP is considered through the enthalpy-pressure relationship. We find that the critical pressure of thermodynamic stability $P_{\text{T}}$ for bulk BP with orthorhombic phase is 4.6 GPa (see Figure S2), very close to the experimental result (4.7 GPa) 29, which indicates a structural phase transition from a A17 (orthorhombic) phase to a A7 (rhombohedral) phase 30. For few-layer phosphorene, a similar structural phase transition is found in accordance to our calculations because of the same mechanism of the structural phase transition as the bulk BP 30. Phonon spectra are then calculated to judge whether or not the thermodynamically stable structures (with $P\leq P_{\text{T}}$) are lattice-dynamic stable. Our results show that all the pressured structures have no imaginary frequencies below $P_{\text{T}}$ 14. The above results ensure that the pressured structures will not experience a structural phase transition below $P_{\text{T}}$; and this is further confirmed by the medium method, showing that the 2D structure is indeed stable and does not favor reconstruction below $P_{\text{T}}$ (see Table S2 and Figure S1).

Topological phase diagram. The layer-pressure topological phase diagram for few-layer phosphorene (below $P_{\text{T}}$, with consideration of SOC) is displayed in Figure 2, where $P_{\text{C}}$ and $P_{\text{T}}$ are the critical pressures for electronic and structural phase transitions, respectively. At $P_{\text{C}}$, an electronic phase transition, from NI to TDSM, takes place (see Figure 3b). TDSM denotes a pristine topological Dirac semimetal phase with only Dirac fermions at $E_{\text{F}}$ (see Figure 3c). No TPTs are observed for $n=2$ and $3$ layers so they are still NIs below $P_{\text{T}}$ (dashed red line), this is because of their large band gaps ($\geq$ 0.75 eV) under zero pressure. With increasing number of $n$, their zero-pressure band gaps gradually decrease, and a NI to TDSM transition happens to occur in the pressure range of P_{\text{C}}$$<$$P$$<$$P_{\text{T}} for $n\geq 4$. We should point out that, although the weak quantum confinement effect herein makes the band inversion occur at a higher pressure $P_{\text{C}}$ when referring to the case in the bulk, this effect could become less important in thicker layers (see Figure S3).

Under pressure-free conditions, 4-layer phosphorene is a NI with a band gap of 0.63 eV as shown in Figure 3a. With increasing pressure, at $P=4.1$ GPa, the valance band maximum (VBM) and conduction band minimum (CBM) touch together at $\Gamma$ point of the Brillouin zone, as typified in Figure 3b. When $P=4.4$ GPa, VBM and CBM with opposite parities have been inverted and a single pair of Dirac cones along $\Gamma-\text{Y}$ (and the opposite direction of $\Gamma-\text{Y}$) are formed (see Figure 2b and Figures 3c, d), implying of a quasi-2D TDSM.

By examining the real space distribution of VBM and CBM at $\Gamma$ point, we find that the two bands are really inversed when P$$>$$P_{\text{C}} (see Figure 4a), supporting the topological phase transition. The $Z_{2}$ topological invariant is calculated to further check whether the inverted band structure is topologically nontrivial or not. The calculation is carried out following the method developed by Fu and Kane 31, based on the fact that inversion symmetry holds for all $n$-layer phosphorene studied here (Table S3). The $Z_{2}$ topological invariant is then obtained from the parity of each pair of Kramers degeneracy occupied band at the time-reversal-invariant momenta (TRIM) points. As shown in Figure 1c, the BZ of $n$-layer phosphorene is a rectangle with four TRIM points: the $\Gamma$ point, the X point, the M point and the Y point. The $Z_{2}$ topological invariant is thus expressed by,

[TABLE]

where $\delta$(Ki) stands for the product of parity eigenvalues at the TRIM points, $\xi=\pm 1$ are the parity eigenvalues and $N$ denotes the number of the degenerated occupied bands. Our results show that the inverted band structure has a non-zero integer $Z_{2}$ topological invariant ($\nu$=1), which ensures a nontrivial topological state.

From 3D mixed Fermions to 2D pure Dirac Fermions. We have reported in an early recent work that bulk BP can convert from a NI into a 3D Dirac semimetal under the hydrostatic pressure, but only a carrier-mixed Dirac state was acquired 13, 14. The obtained 3D Dirac semimetal state displays a node-loop (red dashed circle) with continuous Dirac points around the Z point in BZ when SOC effect is not considered (see Figure 4b), which is in agreement with a previous theoretical study 32. In fact, only two pairs of them (on the loop along Z-M path) are exactly located at $E_{\text{F}}$, while others are located within $\pm$0.15 eV around $E_{\text{F}}$ 14. Even with consideration of SOC, a single pair of Dirac cones along Z-T path cannot be opened up due to the protection of glide-plane symmetry, while others on the loop are opened up with a gap ($\leq$10 meV). In the 2D case, the single pair of Dirac cones are projected to the 2D BZ along the $\bar{{\Gamma}}$-$\bar{\text{Y}}$ path (see Figure 4b). For the pressured bulk BP, it only displays a mixture phase with the combined character of trival semimetals and topological Dirac semimetals (mTTDSM), which shows (hole-type and electron-type) Dirac fermions mixed with normal fermions 13, 14. The mTTDSM phase in bulk BP originates from the stronger anisotropic momentum and the charge compensation in $k_{y}$-$k_{z}$ plane. In contrast, the $n$-layer phosphorene is able to exhibit a pure TDSM phase with a single pair of Dirac points locating exactly at $E_{\text{F}}$ under a certain pressure range, owing to the vanishment of anisotropic momentum $k_{z}$ from the reduced dimensional effect.

Robust Dirac cones under external perturbation. In order to explicitly understand which space group elements (glide plane and/or screw axis) protect the Dirac cones in few-layer phosphorene, we calculate the band structures of the few-layer structure with particular deformations above $P_{\text{C}}$. Three deformations are considered, by moving the outmost P atoms by 0.1 Å along $x$, $y$ and $z$ directions, respectively (see Figure 5). Our results show that translations along $y$ and $z$ directions cannot result in Dirac cones opening (see Figure S4), because the YZ glide plane at the middle of the zigzag chain retains (see Table S4). Therefore, these Dirac cones are protected by the nonsymmorphic space symmetry (out-of-plane glide plane) in quasi-2D phosphorene.

Recently, numerous studies show contradiction about whether the strain-induced Dirac cones in monolayer or few-layer phosphorene can be opened up by SOC 10, 4, 2. To our knowledge, the DFT calculations usually sort different bands according to their magnitude discrepancy, which probably leads to artificial mini-gap in band structures (see Figure S5a). To understand the band structure of crystals, it is very important to sort the bands according to their eigenvector continuity. A band-sorting method, which is based on $k\cdot p$ theory 33, is designed here to sort the DFT wave functions (see the derivation steps in the Supporting Information), i.e.,

[TABLE]

where $n$, $J$ and $\Delta_{k}$ are the band index, total angular momentum and a small wave vector, respectively; $c_{i}(k)$ is the coefficient of the $i$th atomic orbital in the wavefunction. The $k{\cdot}p$ theory has been applied to sort phonon bands, resulting in the discovery of various lattice dynamical mechanisms 33, 34. Such basic algorithm is reformulated here to sort electronic bands, which can help us to efficiently distinguish mini-gap from band crossing. It is especially useful to investigate the complex electronic properties of topological materials. The resorted bands corroborate that the Dirac cones can not be opened up by SOC (see Figure S5b), further confirming that Dirac points are protected by the intrinsic symmetry ($i.e.$ glide plane). Our results are in good agreement with those derived from the model methods 2, 4 with the consideration of the special symmetry elements. Recently, we notice that it remains in conflict between the theoretical models and DFT calculations about whether the Dirac cones induced by external electric field in few-layer phosphorene can be opened up by SOC 35, 8, 2, 3. We then employ our band-sorting methodology to deal with the electric field case. Our results explicitly show that the Dirac cones herein can not be opened up for a finite gap even with the largest field strength reported in literature 35, 8, which can be understood by the fact that the glide-reflection symmetry retains regardless of the perpendicular electric fields 2. Thus, here we elucidate that the Dirac cones are robust in respect to strain, pressure or external electric field, and this rationale is supported by a list of recent theoretial results and experimental observations 2, 3, 4, 14, 9, 13.

In summary, we present in this work that a 2D pure topological Dirac semimetal phase (TDSM) can be feasibly and effectively achieved in few-layer phosphorene by adopting a moderate hydrostatic pressure. The electronic band structure of the 2D TDSM shows a single pair of Dirac points locating exactly at the Fermi level, which are protected by the nonsymmorphic space symmetry (glide plane) and thus can not be opened up by SOC. We also corroborate and clarify that these Dirac cones are robust under the external perturbation (including strain or external electric field), if not breaking the glide-plane symmetry within each layer. The pressure-tunable topological properties of few-layer van der Waals materials may offer great flexibility in design and optimization of electronic and optoelectronic devices.

{suppinfo}

This file contains five parts: computational details, thermodynamic stability, quantum confinement effect, reduced dimensional effect and symmetry protected Dirac cones, and sorting the band dispersions based on k$$\cdot $p$ theory.

{acknowledgement}

The authors thank Dr. Rui Wang, Jin-Zhu Zhao, Xian-Long Wang and Jie Zhang for many helpful discussions on the subject. This work was supported by National key research and development program (Grant No. 2016YFB0901600), the NSF of China under Grant Nos. 11474145, 11334003, 11534010, the Nanshan Key Lab on Nonvolatile Memory Grant (KC2015ZDYF0003A), and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No. U1501501.